The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
This document provides an overview of exponential functions including:
- Definitions of exponential functions for positive integer, rational, and irrational exponents. Conventions are established to define exponents for all real number bases and exponents.
- Properties of exponential functions including that they are continuous functions with domain of all real numbers and range from 0 to infinity. Properties involving addition, subtraction, multiplication, and division of exponents are proven.
- Examples are provided to demonstrate simplifying expressions using properties of exponents, including fractional exponents.
This document discusses functions and graphs. It begins by introducing the concept of a function and how functions are used to model real-world phenomena by relating one quantity to another. Examples are given such as relating distance fallen to time for a falling object. The document then discusses different types of functions like quadratic, polynomial, and rational functions. It provides guidelines for graphing these different function types by identifying intercepts, end behavior, maxima/minima, and asymptotes. The document also covers combining functions through composition and finding inverse functions. It concludes by discussing using functions to model real-world scenarios like relating crop yield to rainfall or fish length to age.
This document discusses how to graph polynomial functions and find local extrema. It provides instructions on making a table of values, plotting points, connecting them with a smooth curve, and checking end behavior based on the degree and leading coefficient. Extrema are defined as local maxima or minima, where the graph changes from increasing to decreasing. Examples are given to demonstrate graphing polynomials and finding turning points that indicate local extrema.
This document discusses trigonometric functions including sine, cosine, tangent, cosecant, secant and cotangent. It provides instructions on finding the exact value of trig functions for various angles in different quadrants using special right triangles and quadrantal angles. It also discusses using a calculator to evaluate trig functions and provides example problems to solve.
The document describes properties of trigonometric functions including sine, cosine, and tangent. It discusses key features of their graphs such as period, amplitude, domain, range, and intercepts. Examples are provided to demonstrate how to sketch the graphs of trigonometric functions using these properties. Key points, periods, and asymptotes are calculated and graphs are drawn.
The document provides an overview of functions including definitions, examples, and properties. It defines a function as a relation that assigns each element in the domain to a single element in the range. Examples of functions expressed by formulas, numerically, graphically, and verbally are given. Properties like monotonicity, symmetry, evenness, and oddness are defined and illustrated with examples. The document aims to introduce the fundamental concepts of functions to readers.
Graphs of the Sine and Cosine Functions LectureFroyd Wess
More: www.PinoyBIX.org
Lesson Objectives
Able to plot the different Trigonometric Graphs
Graph of Sine Function (y = f(x) = sinx)
Graph of Cosine Function (y = f(x) = cosx)
Define the Maximum and Minimum value in a graph
Generalized Trigonometric Functions
Graphs of y = sinbx
Graphs of y = sin(bx + c)
Could find the Period of Trigonometric Functions
Could find the Amplitude of Trigonometric Functions
Variations in the Trigonometric Functions
This document provides an overview of exponential functions including:
- Definitions of exponential functions for positive integer, rational, and irrational exponents. Conventions are established to define exponents for all real number bases and exponents.
- Properties of exponential functions including that they are continuous functions with domain of all real numbers and range from 0 to infinity. Properties involving addition, subtraction, multiplication, and division of exponents are proven.
- Examples are provided to demonstrate simplifying expressions using properties of exponents, including fractional exponents.
This document discusses functions and graphs. It begins by introducing the concept of a function and how functions are used to model real-world phenomena by relating one quantity to another. Examples are given such as relating distance fallen to time for a falling object. The document then discusses different types of functions like quadratic, polynomial, and rational functions. It provides guidelines for graphing these different function types by identifying intercepts, end behavior, maxima/minima, and asymptotes. The document also covers combining functions through composition and finding inverse functions. It concludes by discussing using functions to model real-world scenarios like relating crop yield to rainfall or fish length to age.
This document discusses how to graph polynomial functions and find local extrema. It provides instructions on making a table of values, plotting points, connecting them with a smooth curve, and checking end behavior based on the degree and leading coefficient. Extrema are defined as local maxima or minima, where the graph changes from increasing to decreasing. Examples are given to demonstrate graphing polynomials and finding turning points that indicate local extrema.
This document discusses trigonometric functions including sine, cosine, tangent, cosecant, secant and cotangent. It provides instructions on finding the exact value of trig functions for various angles in different quadrants using special right triangles and quadrantal angles. It also discusses using a calculator to evaluate trig functions and provides example problems to solve.
The document describes properties of trigonometric functions including sine, cosine, and tangent. It discusses key features of their graphs such as period, amplitude, domain, range, and intercepts. Examples are provided to demonstrate how to sketch the graphs of trigonometric functions using these properties. Key points, periods, and asymptotes are calculated and graphs are drawn.
1. The rational function f(x) = 1/x^2 has a vertical asymptote at x = 2 because the denominator is 0 at that point.
2. As x values approach 2 from either side, y values approach positive or negative infinity, respectively.
3. The graph gets closer and closer to the vertical line x = 2 but never touches it. This line is called a vertical asymptote.
4. The graph also has a horizontal asymptote at y = 0, as y values approach 0 as x values increase or decrease indefinitely.
This document discusses radian and degree measure of angles. It covers terminology used to describe angles, converting between radian and degree measure, finding coterminal angles, and classifying angles by quadrant. Radian measure is defined as the measure of an angle whose terminal side intercepts an arc of length r on a circle of radius r. Common conversions between degrees and radians are also provided.
This document discusses trigonometric functions and different ways of measuring angles. It introduces degrees and radians as units of angular measurement. Degrees are divided into minutes and seconds, with one degree equal to 1/360 of a full rotation. Radians are defined as the angle subtended by an arc of a circle whose length is equal to the radius. Some key conversions are provided, such as 1 radian being approximately 57.3 degrees and 2 pi radians equaling 360 degrees. Coterminal angles which share the same initial and terminal sides are also discussed.
An identity is a statement that two trigonometric expressions are equal for every value of the variable. Identities can be verified by manipulating one side of the equation using algebraic substitutions and trigonometric identities until it matches the other side, without moving terms across the equal sign. Practice is important to get better at verifying identities, as each one may require a different approach. Students should keep trying different methods and not get discouraged if it takes time to solve an identity.
1. The differentiation formulas for the trigonometric functions sin(u), cos(u), tan(u), cot(u), sec(u), and csc(u) are presented in terms of their derivatives du/dx.
2. The derivatives are obtained by applying the basic differentiation formulas for sin(u) and cos(u) along with the quotient, product, and chain rules.
3. Formulas are also provided for deriving the derivatives of inverse trigonometric functions like arcsin, arccos, arctan, etc.
This presentation is intended to help the education students by giving an idea on how they will use technology in education specially in teaching mathematics.
The document provides an introduction to the precise definition of a limit in calculus. It begins with a heuristic definition of a limit using an error-tolerance game between two players. It then presents Cauchy's precise definition, where the limit is defined using epsilon-delta relationships such that for any epsilon tolerance around the proposed limit L, there exists a corresponding delta tolerance around the point a such that the function values are within epsilon of L when the input values are within delta of a. Examples are provided to illustrate the definition. Pathologies where limits may not exist are also discussed.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
6.6 analyzing graphs of quadratic functionsJessica Garcia
This document discusses analyzing and graphing quadratic functions. It defines key terms like vertex, axis of symmetry, and vertex form. It explains that the graph of y=ax^2 is a parabola, and how the value of a affects whether the parabola opens up or down. It also describes how to graph quadratic functions in vertex form by plotting the vertex and axis of symmetry, and using symmetry.
This is a simple PowerPoint on the properties of Sine and Cosine functions. It was created for a student teaching lesson that I had in the past. Feel free to use and modify! :-)
This document discusses four methods for graphing linear equations on a coordinate plane:
1. Using any two points on the line.
2. Using the x-intercept and y-intercept.
3. Using the slope and y-intercept.
4. Using the slope and one known point.
Examples are provided to illustrate each method. Graphing linear equations is important for visualizing relationships between variables in real-life situations.
12. Angle of Elevation & Depression.pptxBebeannBuar1
This document discusses angles of elevation and depression. It defines an angle of elevation as the angle formed between a horizontal line and the line of sight to an object located above the horizontal line. An angle of depression is defined as the angle formed between a horizontal line and the line of sight to an object located below the horizontal line. The document provides examples of solving problems involving angles of elevation and depression using trigonometric functions like tangent, sine, and the Pythagorean theorem. It emphasizes drawing a diagram, identifying if it is an angle of elevation or depression, and using the appropriate trigonometric ratio to solve for missing lengths.
This document provides information about trigonometric functions including:
- The objectives are to convert between degrees and radians, recognize trigonometric identities, and solve trigonometric equations.
- Trigonometry has a long history dating back to ancient civilizations for measuring distances and heights. It is now widely used in fields like astronomy.
- It discusses angles, the unit circle, trigonometric ratios, special angle values, identities, conversions between degrees and radians, and solving trigonometric equations.
This document provides an overview of kinematic equations and how to approach solving kinematics problems. It includes:
- A list of common kinematic equations and how they are used depending on whether an object starts from rest, has constant velocity, or constant acceleration.
- Guidance on identifying key variables like displacement, velocity, acceleration and determining the correct signs based on the problem context.
- Worked examples showing how to set up and solve kinematics problems step-by-step using the appropriate equations.
- Tips for dealing with problems that may require using multiple equations or are multi-stage problems where acceleration changes. The key is to solve for available variables and use those solutions to determine missing variables
The document announces that the midterm exam for the calculus class will cover sections 5.1 through 5.6 on integration by parts. It also lists the date for the midterm, an upcoming movie day, the tentative date for the final exam, times for problem sessions, and the professor's office hours.
The document discusses inverse trigonometric functions and how to define their inverses by restricting the domains of the trig functions. It explains that the sine function's inverse is defined on [-1,1] and the cosine function's inverse is defined on [0,π]. Similarly, the tangent function's inverse is defined on (-π/2, π/2). Graphs and examples of the inverse sine, cosine, and tangent functions are provided.
The document discusses relations, functions, domains, and ranges. It defines a relation as a set of ordered pairs and a function as a relation where each x-value is mapped to only one y-value. It explains how to identify the domain and range of a relation, and use the vertical line test and mappings to determine if a relation is a function. Examples of evaluating functions are also provided.
The document discusses different types of functions and their graphs. It provides examples of constant functions, linear functions, quadratic functions, and absolute value functions. It shows how to graph each type of function by plotting points and describes their domains and ranges. For linear functions, the domain is all real numbers and the range is also all real numbers. For quadratic functions, the graph is a parabola and the range only includes positive values. Absolute value functions have a domain of all real numbers and a range that is positive and excludes negative values below the constant.
The document discusses kinematic equations, which describe motion without considering its causes. It presents four equations that can be used to determine unknown values like velocity, acceleration, displacement, and time, given other known values. These equations can model constant velocity or constant acceleration motion. Examples show how to apply the equations to calculate final velocity, distance traveled, initial velocity, and other values in scenarios involving cars, kicked balls, sledding, and road rage.
The document outlines the course descriptions for a Bachelor of Arts in Business Communication degree program. It lists 48 courses across various subject areas including business mathematics, statistics, psychology, literature, science, keyboarding, call center operations, customer service, public relations, management, and business law. The program aims to provide students with both theoretical knowledge and practical skills for a career in business communication. It culminates in a 12-credit practicum or thesis writing requirement where students apply their learning in a professional field.
This document provides a self-introduction that lists the author's occupations as an artist, cultural worker, trainer, organizer, researcher, and creative. It also includes brief sections on personal branding, behavioral patterns inventory, anti-violence programs, and emotional intelligence.
1. The rational function f(x) = 1/x^2 has a vertical asymptote at x = 2 because the denominator is 0 at that point.
2. As x values approach 2 from either side, y values approach positive or negative infinity, respectively.
3. The graph gets closer and closer to the vertical line x = 2 but never touches it. This line is called a vertical asymptote.
4. The graph also has a horizontal asymptote at y = 0, as y values approach 0 as x values increase or decrease indefinitely.
This document discusses radian and degree measure of angles. It covers terminology used to describe angles, converting between radian and degree measure, finding coterminal angles, and classifying angles by quadrant. Radian measure is defined as the measure of an angle whose terminal side intercepts an arc of length r on a circle of radius r. Common conversions between degrees and radians are also provided.
This document discusses trigonometric functions and different ways of measuring angles. It introduces degrees and radians as units of angular measurement. Degrees are divided into minutes and seconds, with one degree equal to 1/360 of a full rotation. Radians are defined as the angle subtended by an arc of a circle whose length is equal to the radius. Some key conversions are provided, such as 1 radian being approximately 57.3 degrees and 2 pi radians equaling 360 degrees. Coterminal angles which share the same initial and terminal sides are also discussed.
An identity is a statement that two trigonometric expressions are equal for every value of the variable. Identities can be verified by manipulating one side of the equation using algebraic substitutions and trigonometric identities until it matches the other side, without moving terms across the equal sign. Practice is important to get better at verifying identities, as each one may require a different approach. Students should keep trying different methods and not get discouraged if it takes time to solve an identity.
1. The differentiation formulas for the trigonometric functions sin(u), cos(u), tan(u), cot(u), sec(u), and csc(u) are presented in terms of their derivatives du/dx.
2. The derivatives are obtained by applying the basic differentiation formulas for sin(u) and cos(u) along with the quotient, product, and chain rules.
3. Formulas are also provided for deriving the derivatives of inverse trigonometric functions like arcsin, arccos, arctan, etc.
This presentation is intended to help the education students by giving an idea on how they will use technology in education specially in teaching mathematics.
The document provides an introduction to the precise definition of a limit in calculus. It begins with a heuristic definition of a limit using an error-tolerance game between two players. It then presents Cauchy's precise definition, where the limit is defined using epsilon-delta relationships such that for any epsilon tolerance around the proposed limit L, there exists a corresponding delta tolerance around the point a such that the function values are within epsilon of L when the input values are within delta of a. Examples are provided to illustrate the definition. Pathologies where limits may not exist are also discussed.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
6.6 analyzing graphs of quadratic functionsJessica Garcia
This document discusses analyzing and graphing quadratic functions. It defines key terms like vertex, axis of symmetry, and vertex form. It explains that the graph of y=ax^2 is a parabola, and how the value of a affects whether the parabola opens up or down. It also describes how to graph quadratic functions in vertex form by plotting the vertex and axis of symmetry, and using symmetry.
This is a simple PowerPoint on the properties of Sine and Cosine functions. It was created for a student teaching lesson that I had in the past. Feel free to use and modify! :-)
This document discusses four methods for graphing linear equations on a coordinate plane:
1. Using any two points on the line.
2. Using the x-intercept and y-intercept.
3. Using the slope and y-intercept.
4. Using the slope and one known point.
Examples are provided to illustrate each method. Graphing linear equations is important for visualizing relationships between variables in real-life situations.
12. Angle of Elevation & Depression.pptxBebeannBuar1
This document discusses angles of elevation and depression. It defines an angle of elevation as the angle formed between a horizontal line and the line of sight to an object located above the horizontal line. An angle of depression is defined as the angle formed between a horizontal line and the line of sight to an object located below the horizontal line. The document provides examples of solving problems involving angles of elevation and depression using trigonometric functions like tangent, sine, and the Pythagorean theorem. It emphasizes drawing a diagram, identifying if it is an angle of elevation or depression, and using the appropriate trigonometric ratio to solve for missing lengths.
This document provides information about trigonometric functions including:
- The objectives are to convert between degrees and radians, recognize trigonometric identities, and solve trigonometric equations.
- Trigonometry has a long history dating back to ancient civilizations for measuring distances and heights. It is now widely used in fields like astronomy.
- It discusses angles, the unit circle, trigonometric ratios, special angle values, identities, conversions between degrees and radians, and solving trigonometric equations.
This document provides an overview of kinematic equations and how to approach solving kinematics problems. It includes:
- A list of common kinematic equations and how they are used depending on whether an object starts from rest, has constant velocity, or constant acceleration.
- Guidance on identifying key variables like displacement, velocity, acceleration and determining the correct signs based on the problem context.
- Worked examples showing how to set up and solve kinematics problems step-by-step using the appropriate equations.
- Tips for dealing with problems that may require using multiple equations or are multi-stage problems where acceleration changes. The key is to solve for available variables and use those solutions to determine missing variables
The document announces that the midterm exam for the calculus class will cover sections 5.1 through 5.6 on integration by parts. It also lists the date for the midterm, an upcoming movie day, the tentative date for the final exam, times for problem sessions, and the professor's office hours.
The document discusses inverse trigonometric functions and how to define their inverses by restricting the domains of the trig functions. It explains that the sine function's inverse is defined on [-1,1] and the cosine function's inverse is defined on [0,π]. Similarly, the tangent function's inverse is defined on (-π/2, π/2). Graphs and examples of the inverse sine, cosine, and tangent functions are provided.
The document discusses relations, functions, domains, and ranges. It defines a relation as a set of ordered pairs and a function as a relation where each x-value is mapped to only one y-value. It explains how to identify the domain and range of a relation, and use the vertical line test and mappings to determine if a relation is a function. Examples of evaluating functions are also provided.
The document discusses different types of functions and their graphs. It provides examples of constant functions, linear functions, quadratic functions, and absolute value functions. It shows how to graph each type of function by plotting points and describes their domains and ranges. For linear functions, the domain is all real numbers and the range is also all real numbers. For quadratic functions, the graph is a parabola and the range only includes positive values. Absolute value functions have a domain of all real numbers and a range that is positive and excludes negative values below the constant.
The document discusses kinematic equations, which describe motion without considering its causes. It presents four equations that can be used to determine unknown values like velocity, acceleration, displacement, and time, given other known values. These equations can model constant velocity or constant acceleration motion. Examples show how to apply the equations to calculate final velocity, distance traveled, initial velocity, and other values in scenarios involving cars, kicked balls, sledding, and road rage.
The document outlines the course descriptions for a Bachelor of Arts in Business Communication degree program. It lists 48 courses across various subject areas including business mathematics, statistics, psychology, literature, science, keyboarding, call center operations, customer service, public relations, management, and business law. The program aims to provide students with both theoretical knowledge and practical skills for a career in business communication. It culminates in a 12-credit practicum or thesis writing requirement where students apply their learning in a professional field.
This document provides a self-introduction that lists the author's occupations as an artist, cultural worker, trainer, organizer, researcher, and creative. It also includes brief sections on personal branding, behavioral patterns inventory, anti-violence programs, and emotional intelligence.
The document summarizes the top 10 HR lessons from 2012 according to various contributing authors to the HR Gazette website. It lists the top 10 lessons such as an employer's potential liability for harassment by independent contractors, the discoverability of social media posts in employment lawsuits, and whether FMLA damages are considered taxable payroll. It then continues listing thank you messages and contributing authors to the HR Gazette.
How To Give A Written Warning to an Employee (With Form)Mary Wright
A written warning is a formal disciplinary action that identifies an employee's observed performance deficiencies or misconduct. It notifies the employee that their performance does not meet expectations and there could be disciplinary consequences if improvement is not made. A written warning should be delivered in a meeting with the employee, their supervisor, a witness, and ideally an HR representative. It is documented and signed by all parties and stored in the employee's personnel file.
The document discusses laws protecting disabled workers, including the Vocational Rehabilitation Act of 1973, Americans with Disabilities Act of 1990, and Americans with Disabilities Act Amendments Act of 2008. It also summarizes a case involving Ford Motor Company terminating an employee with irritable bowel syndrome, claiming she was a subpar performer who refused reasonable accommodations, while she claimed discrimination. The court found in favor of Ford, as they had policies for accommodating disabilities and the employee did not qualify as protected under the ADA.
What is an "Essential Job Function" (under workplace disability laws)Mary Wright
The document discusses the definition of "essential job functions" under the Americans with Disabilities Act (ADA) and California Fair Employment and Housing Act (FEHA). Both laws prohibit disability discrimination and require employers to provide reasonable accommodations. The laws also require employers to determine the essential functions of each job in order to prepare job descriptions, handle accommodation requests, and determine if accommodation is needed. The document defines essential functions as fundamental duties that the position exists to perform or that are highly specialized. It provides examples of evidence to consider a function essential and notes that marginal functions are not essential.
This document provides guidance to managers on addressing employee performance problems through progressive discipline. It discusses the concept of progressive discipline and includes verbal counseling, written warnings, letters of reprimand, suspension, demotion and dismissal. Sample letters are also provided for each step of progressive discipline.
This document provides a template and sample letters for supervisors to use when issuing disciplinary notices to employees for misconduct such as insubordination, excessive absenteeism, or other unprofessional behavior. The template outlines the necessary components of disciplinary letters and notes that human resources should review disciplinary actions prior to implementation.
Lesson 20: Derivatives and the Shape of Curves (Section 041 slides)Mel Anthony Pepito
Describe the monotonicity of f(x) = arctan(x).
The derivative of arctan(x) is f'(x) = 1/(1+x^2), which is always positive. Therefore, f(x) = arctan(x) is an increasing function on (-∞,∞).
Lesson 19: The Mean Value Theorem (Section 021 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Lesson 19: The Mean Value Theorem (Section 041 handout)Matthew Leingang
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Lesson20 -derivatives_and_the_shape_of_curves_021_slidesMel Anthony Pepito
f is decreasing on (−∞, −4/5] and increasing on [−4/5, ∞).
. . . . . .
V63.0121.021, Calculus I (NYU) Section 4.2 The Shapes of Curves November 16, 2010 13 / 32
This document is from a Calculus I class at New York University. It provides an overview of functions, including the definition of a function, different ways functions can be represented (formulas, tables, graphs, verbal descriptions), properties of functions like monotonicity and symmetry, and examples of determining domains and ranges of functions. It aims to help students understand functions and their representations as a foundation for calculus.
No, this is not a function because the same input of 2 is mapped to two different outputs of 4 and 5. For a relation to be a function, each input must map to a unique output.
V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
Lesson 19: The Mean Value Theorem (Section 021 slides)Matthew Leingang
(a) E-ZPass cannot prove that the driver was speeding. E-ZPass records entry and exit times and locations, but does not continuously track speed. It cannot determine the driver's exact speed at any point during the trip, so it cannot prove a specific speeding violation occurred. The best it could show is an average speed that may or may not indicate speeding depending on the specific speed limit(s) along the route.
Lesson 13: Exponential and Logarithmic Functions (Section 041 slides)Mel Anthony Pepito
This document contains lecture notes on exponential and logarithmic functions from a Calculus I class at New York University. The notes cover definitions of exponential functions, properties of exponential functions including laws of exponents, the natural exponential function e, and logarithmic functions. The objectives are to understand exponential functions, their properties, and laws of logarithms including the change of base formula.
Lesson 13: Exponential and Logarithmic Functions (Section 041 slides)Matthew Leingang
This document contains lecture notes on exponential and logarithmic functions from a Calculus I class at New York University. The notes cover definitions of exponential functions, properties of exponential functions including laws of exponents, the natural exponential function e, and logarithmic functions. The objectives are to understand exponential functions, their properties, and laws of logarithms including the change of base formula.
Lesson 13: Exponential and Logarithmic Functions (Section 021 slides)Mel Anthony Pepito
This document contains lecture notes from a Calculus I class at New York University on October 21, 2010. It discusses the objectives and outline for sections 3.1-3.2 on exponential and logarithmic functions. Key points include definitions of exponential functions, properties like ax+y = axay, and conventions for extending exponents to non-positive integers and irrational numbers.
Lesson 2: A Catalog of Essential Functions (handout)Matthew Leingang
The document outlines topics to be covered in a Calculus I class, including essential functions such as linear, polynomial, rational, trigonometric, exponential, and logarithmic functions. It provides examples and definitions of these functions and notes how their graphs can be transformed through vertical and horizontal shifts. Assignments include WebAssign problems due January 31st and a written assignment due February 2nd.
Lesson 20: Derivatives and the Shape of Curves (Section 041 handout)Matthew Leingang
1) The document discusses using derivatives to determine the shapes of curves, including intervals of increasing/decreasing behavior, local extrema, and concavity.
2) Key tests covered are the increasing/decreasing test, first derivative test, concavity test, and second derivative test. These allow determining monotonicity, local extrema, and concavity from the signs of the first and second derivatives.
3) Examples demonstrate applying these tests to find the intervals of monotonicity, points of local extrema, and intervals of concavity for various functions.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It provides objectives for understanding rules like the derivative of a constant function, the constant multiple rule, sum rule, and derivatives of sine and cosine functions. It then gives examples of finding the derivative of a squaring function using the definition, and introduces the concept of the second derivative.
This document is a section from a Calculus I course at New York University that discusses basic differentiation rules. It covers objectives like differentiating constant, sum, and difference functions. It also reviews derivatives of sine and cosine. Examples are provided, like finding the derivative of a squaring function x^2 using the definition of a derivative. The document outlines the topics and provides explanations and step-by-step solutions.
This document discusses the definite integral and its properties. It begins by stating the objectives of computing definite integrals using Riemann sums and limits, estimating integrals using approximations like the midpoint rule, and reasoning about integrals using their properties. The outline then reviews the integral as a limit of Riemann sums and how to estimate integrals. It also discusses properties of the integral and comparison properties. Finally, it restates the theorem that if a function is continuous, the limit of Riemann sums is the same regardless of the choice of sample points.
The document discusses the definite integral, including computing it using Riemann sums, estimating it using approximations like the midpoint rule, and reasoning about its properties. It outlines the topics to be covered, such as recalling previous concepts and comparing properties of integrals. Formulas are provided for calculating Riemann sums using different representative points within the intervals.
This document contains lecture notes on differentiation rules from a Calculus I class at New York University. It begins with objectives to understand basic differentiation rules like the derivative of a constant function, the Constant Multiple Rule, the Sum Rule, and derivatives of sine and cosine. It then provides examples of using the definition of the derivative to find the derivatives of squaring and cubing functions. It illustrates the functions and their derivatives on graphs and discusses properties like a function being increasing when its derivative is positive.
Similar to Lesson 2: A Catalog of Essential Functions (20)
This document provides guidance on developing effective lesson plans for calculus instructors. It recommends starting by defining specific learning objectives and assessments. Examples should be chosen carefully to illustrate concepts and engage students at a variety of levels. The lesson plan should include an introductory problem, definitions, theorems, examples, and group work. Timing for each section should be estimated. After teaching, the lesson can be improved by analyzing what was effective and what needs adjustment for the next time. Advanced preparation is key to looking prepared and ensuring students learn.
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
This document discusses electronic grading of paper assessments using PDF forms. Key points include:
- Various tools for creating fillable PDF forms using LaTeX packages or desktop software.
- Methods for stamping completed forms onto scanned documents including using pdftk or overlaying in TikZ.
- Options for grading on tablets or desktops including GoodReader, PDFExpert, Adobe Acrobat.
- Extracting data from completed forms can be done in Adobe Acrobat or via command line with pdftk.
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
Lesson 27: Integration by Substitution (handout)Matthew Leingang
This document contains lecture notes on integration by substitution from a Calculus I class. It introduces the technique of substitution for both indefinite and definite integrals. For indefinite integrals, the substitution rule is presented, along with examples of using substitutions to evaluate integrals involving polynomials, trigonometric, exponential, and other functions. For definite integrals, the substitution rule is extended and examples are worked through both with and without first finding the indefinite integral. The document emphasizes that substitution often simplifies integrals and makes them easier to evaluate.
Lesson 26: The Fundamental Theorem of Calculus (handout)Matthew Leingang
1) The document discusses lecture notes on Section 5.4: The Fundamental Theorem of Calculus from a Calculus I course. 2) It covers stating and explaining the Fundamental Theorems of Calculus and using the first fundamental theorem to find derivatives of functions defined by integrals. 3) The lecture outlines the first fundamental theorem, which relates differentiation and integration, and gives examples of applying it.
This document contains notes from a calculus class lecture on evaluating definite integrals. It discusses using the evaluation theorem to evaluate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. The document also contains examples of evaluating definite integrals, properties of integrals, and an outline of the key topics covered.
This document contains lecture notes from a Calculus I class covering Section 5.3 on evaluating definite integrals. The notes discuss using the Evaluation Theorem to calculate definite integrals, writing derivatives as indefinite integrals, and interpreting definite integrals as the net change of a function over an interval. Examples are provided to demonstrate evaluating definite integrals using the midpoint rule approximation. Properties of integrals such as additivity and the relationship between definite and indefinite integrals are also outlined.
Lesson 24: Areas and Distances, The Definite Integral (handout)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Lesson 24: Areas and Distances, The Definite Integral (slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document contains lecture notes from a Calculus I class discussing optimization problems. It begins with announcements about upcoming exams and courses the professor is teaching. It then presents an example problem about finding the rectangle of a fixed perimeter with the maximum area. The solution uses calculus techniques like taking the derivative to find the critical points and determine that the optimal rectangle is a square. The notes discuss strategies for solving optimization problems and summarize the key steps to take.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
The document discusses curve sketching of functions by analyzing their derivatives. It provides:
1) A checklist for graphing a function which involves finding where the function is positive/negative/zero, its monotonicity from the first derivative, and concavity from the second derivative.
2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
The document contains lecture notes on curve sketching from a Calculus I class. It discusses using the first and second derivative tests to determine properties of a function like monotonicity, concavity, maxima, minima, and points of inflection in order to sketch the graph of the function. It then provides an example of using these tests to sketch the graph of the cubic function f(x) = 2x^3 - 3x^2 - 12x.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
Lesson 20: Derivatives and the Shapes of Curves (handout)Matthew Leingang
This document contains lecture notes on calculus from a Calculus I course. It covers determining the monotonicity of functions using the first derivative test. Key points include using the sign of the derivative to determine if a function is increasing or decreasing over an interval, and using the first derivative test to classify critical points as local maxima, minima, or neither. Examples are provided to demonstrate finding intervals of monotonicity for various functions and applying the first derivative test.
Lesson 20: Derivatives and the Shapes of Curves (handout)
Lesson 2: A Catalog of Essential Functions
1. Section 2.2
A Catalogue of Essential Functions
V63.0121.021/041, Calculus I
New York University
September 9, 2010
Announcements
First WebAssign-ments are due September 14
First written assignment is due September 16
Do the Get-to-Know-You survey (on Blackboard) and Photo for
extra credit!
. . . . . .
2. Announcements
First WebAssign-ments are
due September 14
First written assignment is
due September 16
Do the Get-to-Know-You
survey (on Blackboard)
and Photo for extra credit!
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 2 / 27
3. Objectives: A Catalog of Essential Functions
Identify different classes of
algebraic functions,
including polynomial
(linear, quadratic, cubic,
etc.), polynomial
(especially linear,
quadratic, and cubic),
rational, power,
trigonometric, and
exponential functions.
Understand the effect of
algebraic transformations
on the graph of a function.
Understand and compute
the composition of two
functions. . . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 3 / 27
4. What is a function?
Definition
A function f is a relation which assigns to to every element x in a set D
a single element f(x) in a set E.
The set D is called the domain of f.
The set E is called the target of f.
The set { y | y = f(x) for some x } is called the range of f.
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 4 / 27
5. Classes of Functions
linear functions, defined by slope an intercept, point and point, or
point and slope.
quadratic functions, cubic functions, power functions, polynomials
rational functions
trigonometric functions
exponential/logarithmic functions
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 5 / 27
6. Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Transcendental Functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 6 / 27
7. Linear functions
Linear functions have a constant rate of growth and are of the form
f(x) = mx + b.
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 7 / 27
8. Linear functions
Linear functions have a constant rate of growth and are of the form
f(x) = mx + b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Write
the fare f(x) as a function of distance x traveled.
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 7 / 27
9. Linear functions
Linear functions have a constant rate of growth and are of the form
f(x) = mx + b.
Example
In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Write
the fare f(x) as a function of distance x traveled.
Answer
If x is in miles and f(x) in dollars,
f(x) = 2.5 + 2x
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 7 / 27
10. Example
Biologists have noticed that the chirping rate of crickets of a certain
species is related to temperature, and the relationship appears to be
very nearly linear. A cricket produces 113 chirps per minute at 70 ◦ F
and 173 chirps per minute at 80 ◦ F.
(a) Write a linear equation that models the temperature T as a function
of the number of chirps per minute N.
(b) What is the slope of the graph? What does it represent?
(c) If the crickets are chirping at 150 chirps per minute, estimate the
temperature.
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 8 / 27
12. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equation
y − y0 = m(x − x0 )
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 9 / 27
13. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 9 / 27
14. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
So an equation for T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 9 / 27
15. Solution
The point-slope form of the equation for a line is appropriate
here: If a line passes through (x0 , y0 ) with slope m, then the line
has equation
y − y0 = m(x − x0 )
80 − 70 10 1
The slope of our line is = =
173 − 113 60 6
So an equation for T and N is
1 1 113
T − 70 = (N − 113) =⇒ T = N − + 70
6 6 6
37
If N = 150, then T = + 70 = 76 1 ◦ F
6
6
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 9 / 27
16. Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Transcendental Functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 10 / 27
17. Quadratic functions take the form
f(x) = ax2 + bx + c
The graph is a parabola which opens upward if a > 0, downward if
a < 0.
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 11 / 27
18. Quadratic functions take the form
f(x) = ax2 + bx + c
The graph is a parabola which opens upward if a > 0, downward if
a < 0.
Cubic functions take the form
f(x) = ax3 + bx2 + cx + d
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 11 / 27
19. Other power functions
Whole number powers: f(x) = xn .
1
negative powers are reciprocals: x−3 = 3 .
x
√
fractional powers are roots: x1/3 = 3 x.
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 12 / 27
20. Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Transcendental Functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 13 / 27
21. Rational functions
Definition
A rational function is a quotient of polynomials.
Example
x3 (x + 3)
The function f(x) = is rational.
(x + 2)(x − 1)
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 14 / 27
22. Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Transcendental Functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 15 / 27
23. Trigonometric Functions
Sine and cosine
Tangent and cotangent
Secant and cosecant
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 16 / 27
24. Exponential and Logarithmic functions
exponential functions (for example f(x) = 2x )
logarithmic functions are their inverses (for example f(x) = log2 (x))
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 17 / 27
25. Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Transcendental Functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 18 / 27
26. Transformations of Functions
Take the squaring function and graph these transformations:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 19 / 27
27. Transformations of Functions
Take the squaring function and graph these transformations:
y = (x + 1)2
y = (x − 1)2
y = x2 + 1
y = x2 − 1
Observe that if the fiddling occurs within the function, a transformation
is applied on the x-axis. After the function, to the y-axis.
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 19 / 27
28. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units
y = f(x) − c, shift the graph of y = f(x) a distance c units
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 20 / 27
29. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units upward
y = f(x) − c, shift the graph of y = f(x) a distance c units
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 20 / 27
30. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units upward
y = f(x) − c, shift the graph of y = f(x) a distance c units downward
y = f(x − c), shift the graph of y = f(x) a distance c units
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 20 / 27
31. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units upward
y = f(x) − c, shift the graph of y = f(x) a distance c units downward
y = f(x − c), shift the graph of y = f(x) a distance c units to the right
y = f(x + c), shift the graph of y = f(x) a distance c units
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 20 / 27
32. Vertical and Horizontal Shifts
Suppose c > 0. To obtain the graph of
y = f(x) + c, shift the graph of y = f(x) a distance c units upward
y = f(x) − c, shift the graph of y = f(x) a distance c units downward
y = f(x − c), shift the graph of y = f(x) a distance c units to the right
y = f(x + c), shift the graph of y = f(x) a distance c units to the left
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 20 / 27
33. Now try these
y = sin (2x)
y = 2 sin (x)
y = e−x
y = −ex
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 21 / 27
34. Scaling and flipping
To obtain the graph of
y = f(c · x), scale the graph of f by c
y = c · f(x), scale the graph of f by c
If |c| < 1, the scaling is a
If c < 0, the scaling includes a
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 22 / 27
35. Scaling and flipping
To obtain the graph of
y = f(c · x), scale the graph of f horizontally by c
y = c · f(x), scale the graph of f by c
If |c| < 1, the scaling is a
If c < 0, the scaling includes a
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 22 / 27
36. Scaling and flipping
To obtain the graph of
y = f(c · x), scale the graph of f horizontally by c
y = c · f(x), scale the graph of f vertically by c
If |c| < 1, the scaling is a
If c < 0, the scaling includes a
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 22 / 27
37. Scaling and flipping
To obtain the graph of
y = f(c · x), scale the graph of f horizontally by c
y = c · f(x), scale the graph of f vertically by c
If |c| < 1, the scaling is a compression
If c < 0, the scaling includes a
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 22 / 27
38. Scaling and flipping
To obtain the graph of
y = f(c · x), scale the graph of f horizontally by c
y = c · f(x), scale the graph of f vertically by c
If |c| < 1, the scaling is a compression
If c < 0, the scaling includes a flip
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 22 / 27
39. Outline
Linear functions
Other Polynomial functions
Other power functions
Rational functions
Transcendental Functions
Trigonometric Functions
Exponential and Logarithmic functions
Transformations of Functions
Compositions of Functions
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 23 / 27
40. Composition is a compounding of functions in
succession
g
. ◦f
.
x
. f
. . g
. . g ◦ f)(x)
(
f
.(x)
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 24 / 27
41. Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 25 / 27
42. Composing
Example
Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f.
Solution
f ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the same.
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 25 / 27
43. Decomposing
Example
√
Express x2 − 4 as a composition of two functions. What is its
domain?
Solution
√
We can write the expression as f ◦ g, where f(u) = u and
g(x) = x2 − 4. The range of g needs to be within the domain of f. To
insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2.
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 26 / 27
44. Summary
There are many classes of algebraic functions
Algebraic rules can be used to sketch graphs
. . . . . .
V63.0121.021/041, Calculus I (NYU) Section 2.2 Essential Functions September 9, 2010 27 / 27